On a Generalized Oscillator System: Interbasis Expansions

This article deals with a nonrelativistic quantum mechanical study of a dynamical system which generalizes the isotropic harmonic oscillator system in three dimensions. The problem of interbasis expansions of the wavefunctions is completely solved. A connection between the generalized oscillator system (projected on the z-line) and the Morse system (in one dimension) is discussed.Comment: 23 pages, Latex File, to be published in International Journal of Quantum Chemistr

Coulomb-oscillator duality in spaces of constant curvature

In this paper we construct generalizations to spheres of the well known Levi-Civita, Kustaanheimo-Steifel and Hurwitz regularizing transformations in Euclidean spaces of dimensions 2, 3 and 5. The corresponding classical and quantum mechanical analogues of the Kepler-Coulomb problem on these spheres are discussed.Comment: 33 pages, LaTeX fil

Studying Turbulence from Doppler-broadened Absorption Lines: Statistics of Logarithms of Intensity

We continue our work on developing techniques for studying turbulence with spectroscopic data. We show that Doppler-broadened absorption spectral lines, in particularly, saturated absorption lines, can be used within the framework of the earlier-introduced technique termed the Velocity Coordinate spectrum (VCS). The VCS relates the statistics of fluctuations along the velocity coordinate to the statistics of turbulence, thus it does not require spatial coverage by sampling directions in the plane of the sky. We consider lines with different degree of absorption and show that for lines of optical depth less than one, our earlier treatment of the VCS developed for spectral emission lines is applicable, if the optical depth is used instead of intensity. This amounts to correlating the logarithms of absorbed intensities. For larger optical depths and saturated absorption lines, we show, that the amount of information that one can use is, inevitably, limited by noise. In practical terms, this means that only wings of the line are available for the analysis. In terms of the VCS formalism, this results in introducing an additional window, which size decreases with the increase of the optical depth. As a result, strongly saturated absorption lines carry the information only about the small scale turbulence. Nevertheless, the contrast of the fluctuations corresponding to the small scale turbulence increases with the increase of the optical depth, which provides advantages for studying turbulence combining lines with different optical depths. Combining different absorption lines one can tomography turbulence in the interstellar gas in all its complexity.Comment: 12 pages, 2 figure

The Cosmic Microwave Background & Inflation, Then & Now

Boomerang, Maxima, DASI, CBI and VSA significantly increase the case for accelerated expansion in the early universe (the inflationary paradigm) and at the current epoch (dark energy dominance), especially when combined with data on high redshift supernovae (SN1) and large scale structure (LSS). There are ``7 pillars of Inflation'' that can be shown with the CMB probe, and at least 5, and possibly 6, of these have already been demonstrated in the CMB data: (1) a large scale gravitational potential; (2) acoustic peaks/dips; (3) damping due to shear viscosity; (4) a Gaussian (maximally random) distribution; (5) secondary anisotropies; (6) polarization. A 7th pillar, anisotropies induced by gravity wave quantum noise, could be too small. A minimal inflation parameter set, \omega_b,\omega_{cdm}, \Omega_{tot}, \Omega_Q,w_Q,n_s,\tau_C, \sigma_8}, is used to illustrate the power of the current data. We find the CMB+LSS+SN1 data give \Omega_{tot} =1.00^{+.07}_{-.03}, consistent with (non-baroque) inflation theory. Restricting to \Omega_{tot}=1, we find a nearly scale invariant spectrum, n_s =0.97^{+.08}_{-.05}. The CDM density, \Omega_{cdm}{\rm h}^2 =.12^{+.01}_{-.01}, and baryon density, \Omega_b {\rm h}^2 = >.022^{+.003}_{-.002}, are in the expected range. (The Big Bang nucleosynthesis estimate is 0.019\pm 0.002.) Substantial dark (unclustered) energy is inferred, \Omega_Q \approx 0.68 \pm 0.05, and CMB+LSS \Omega_Q values are compatible with the independent SN1 estimates. The dark energy equation of state, crudely parameterized by a quintessence-field pressure-to-density ratio w_Q, is not well determined by CMB+LSS (w_Q < -0.4 at 95% CL), but when combined with SN1 the resulting w_Q < -0.7 limit is quite consistent with the w_Q=-1 cosmological constant case.Comment: 20 pages, 8 figures, in Theoretical Physics, MRST 2002: A Tribute to George Libbrandt (AIP), eds. V. Elias, R. Epp, R. Myer

Gravitational instability in the strongly nonlinear regime: A study of various approximations

We study the development of gravitational instability in the strongly non-linear regime. For this purpose we use a number of statistical indicators such as filamentary statistics, spectrum of overdense/underdense regions and the void probability function, each of which probes a particular aspect of gravitational clustering. We use these statistical indicators to discriminate between different approximations to gravitational instability which we test against N-body simulations. The approximations which we test are, the truncated Zel'dovich approximation (TZ), the adhesion model (AM), and the frozen flow (FF) and linear potential (LP) approximations. Of these we find that FF and LP break down relatively early, soon after the non-linear length scale exceeds $R_*$ -- the mean distance between peaks of the gravitational potential. The reason for this break down is easy to understand, particles in FF are constrained to follow the streamlines of the initial velocity field. Shell crossing is absent in this case and structure gradually freezes as particles begin to collect near minima of the gravitational potential. In LP particles follow the lines of force of the primordial potential, oscillating about its minima at late times when the non-linear length scale $k_{\rm NL}^{-1}\simeq R_*$. Unlike FF and LP the adhesion model (and to some extent TZ) continues to give accurate results even at late times when $k_{\rm NL}^{-1} \ge R_*$. This is because both AM and TZ useComment: mn.sty (Latex), 20 pages + 11 figures (not included: hardcopy available on request from [email protected]), Submitted to MNRAS, IUCAA-24/9

Quantum Systems with Hidden Symmetry. Interbasis Expansions

This monograph is the English version of the book "Quantum systems with hidden symmetry. Interbasis expansions" published in 2006 by the publishing house FIZMATLIT (Moscow) in Russian. When compiling this version of the book, typos and inaccuracies noted since the release of the Russian edition have been corrected

Families of classical subgroup separable superintegrable systems

We describe a method for determining a complete set of integrals for a classical Hamiltonian that separates in orthogonal subgroup coordinates. As examples, we use it to determine complete sets of integrals, polynomial in the momenta, for some families of generalized oscillator and Kepler-Coulomb systems, hence demonstrating their superintegrability. The latter generalizes recent results of Verrier and Evans, and Rodriguez, Tempesta and Winternitz. Another example is given of a superintegrable system on a non-conformally flat space.Comment: 9 page